Quick take: Democratic president + Republican House + polarization = shutdown

 That's just math.  In fact, if you want a formal proof of the arithmetic, I went through the rigamarole.  It was in my second book, Incremental Polarization: A Unified Spatial Theory of Legislative Elections, Parties & Roll Call Voting.  The short version is this.  Most roll call votes pit a status quo point against a specific alternative.  A representative votes yes if he is closer to the alternative, and no if he is closer to the status quo.  Most status quo points are within a particular range of the middle because it is hard to get policy away from the floor median, and easier to move policy towards the floor median.  Hence, if you control the agenda and have noncentrist preferences, often your best move is keeping alternatives from reaching the floor.

But.

There are also these things called "reversion points."  Those are policies to which we revert when nothing passes, which are different from the status quo.  Fail to pass something, and a change occurs.  These are most important in the case of a debt ceiling or spending bills.  Pass nothing in the case of the former, and we default instead of making payments.  Pass nothing in the case of the latter, and we get a shutdown.

These reversion points are ideologically extreme.  You know who likes these ideologically extreme reversion points?  The right flank of the Republican Party.  In the case of the debt ceiling, GOP leadership is so terrified of not just what the default would mean to their own, personal pocketbooks, but of taking the blame for the nationwide economic calamity that they won't let the wingnuts get their way.  A shutdown, though?  The cost of a temporary shutdown is low enough that GOP House leadership lets them have their way, at least for a time.

Is this stupid?  No.  It is magnificently, epically, colossally stupid.  However, when a large faction of the majority party in the House kinda likes shutdowns, because a shutdown is a reversion point close to their ideal points, they won't budge very far to prevent it.  Ask them to compromise at all with a Democratic president to prevent this thing that they like, and they'll tell you to go fuck yourself.

Hence, Democratic president + Republican House + polarization = shutdown.

For the full math, go read my 2018 book, Incremental Polarization.  It will also tell you that McCarthy's days are probably numbered.  This is what got Boehner.

Jardin De La Croix, "Reversion," from Circadia.